A geometrical version of the higher order Hamilton formalism in fibred manifolds

Abstract

It has been clarified recently that an r-th order Lagrangian on a fibred manifold Y → X does not determine a unique Poincaré-Cartan form provided dim X > 1 and r > 2, [1], [4], [6], [9], [10]. To make this fact more transparent, we introduced a new operation generalizing the formal exterior differentiation, [6]. In the present paper we deduce in such a way that a unique Poincaré-Cartan form can be determined by means of a simple additional structure - a linear symmetric connection Г on the base manifold X (or, more generally, by a convenient splitting S). Then we present a suitable geometric definition of a regular r-th order Lagrangian on Y and we prove that any our Poincaré-Cartan form can be used in a geometrical version of the higher order Hamilton formalism.